Finite Group Databasebeta

← #10594591

Subgroups — #10594591

Order 576.

426 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64097C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic, complemented
3C3×#896C211yesmaximal, characteristic, complemented
4#63913C211yesmaximal, characteristic
5#63905C211yesmaximal, characteristic
6#63896C211yesmaximal, characteristic, complemented
7#63891C211yesmaximal, characteristic
8#63887C211yesmaximal, characteristic, complemented
9#63866C211yesmaximal, characteristic, complemented
10#63862C211yesmaximal, characteristic, complemented
11#63862C211yesmaximal, characteristic, complemented
12#63859C211yesmaximal, characteristic, complemented
13#63857C211yesmaximal, characteristic, complemented
14#63846C211yesmaximal, characteristic, complemented
15#63845C211yesmaximal, characteristic, complemented
16#5712—31nomaximal
17#5646—31nomaximal
18C2×#634C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×C3×Dic6C2211yescharacteristic
22C2×#625C2211yescharacteristic
23C2×D3×Dic3C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26#3750C2211yescharacteristic
27#3750C2211yescharacteristic
28C4×C3⋊Dic3C2211yescharacteristic
29C3×(C2×D3)⋊C4C2211yescharacteristic
30C3×(C2×D3)⋊C4C2211yescharacteristic
31C3×(C2×D3)⋊C4C2211yescharacteristic
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×C4⋊Dic3C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×C4×Dic3C2211yescharacteristic
39#3724C2211yescharacteristic
40Dic3⋊Dic3C2211yescharacteristic
41Dic3⋊Dic3C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#126—31no—
54C3×C22⋊Q8—31no—
55#946—31no—
56#944—31no—
57#929—31no—
58#924—31no—
59#898—31no—
60#898—31no—
61#898D311yescharacteristic, complemented
62#897—31no—
63#896—31no—
64#896D311yescharacteristic, complemented
65#890—31no—
66#890—31no—
67#889—31no—
68#889—31no—
69#888—31no—
70#888—31no—
71#888—61no—
72#886—31no—
73#885—31no—
74#885—31no—
75#883—31no—
76#883—31no—
77#879—31no—
78#879—31no—
79#879—31no—
80#878—31no—
81#878—31no—
82#876—31no—
83#869—31no—
84C22×C3×D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93#634—41no—
94#634—41no—
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C3×C4×D3—41no—
102C3×Dic6—41no—
103#625—41no—
104D3×Dic3—41no—
105#563—91noSylow
106C2×C3×Q8—31no—
107C22×C3×C4—31no—
108C2×#67—31no—
109C2×#67—31no—
110C22×Dic3—31no—
111C2×C4×D3—31no—
112C2×C4×D3C2×D311yescharacteristic
113C2×Dic6—31no—
114C2×Dic6—31no—
115C2×Dic6—31no—
116C2×Dic6—31no—
117C2×Dic6—61no—
118C2×Dic6—61no—
119C2×Dic6C2×D311yescharacteristic
120C3×C4⋊C4—21no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4C2×D311yescharacteristic
126C3×C4⋊C4C2×D311yescharacteristic
127C3×C22⋊C4—31no—
128C3×C22⋊C4—31no—
129C3×C22⋊C4—31no—
130C3×C22⋊C4—31no—
131C3×C22⋊C4—31no—
132C3×C42—31no—
133#217—31no—
134#217—31no—
135#217—31no—
136#217—31no—
137#217—31no—
138(C2×D3)⋊C4—31no—
139(C2×D3)⋊C4—31no—
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4—31no—
143(C2×D3)⋊C4C2×D311yescharacteristic
144(C2×D3)⋊C4C2×D311yescharacteristic
145(C2×D3)⋊C4C2×D311yescharacteristic
146(C2×D3)⋊C4C2×D311yescharacteristic
147(C2×D3)⋊C4C2×D311yescharacteristic
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—61no—
158C4⋊Dic3C2×D311yescharacteristic
159Dic3⋊C4—31no—
160Dic3⋊C4—31no—
161Dic3⋊C4—31no—
162Dic3⋊C4—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—31no—
171Dic3⋊C4—31no—
172Dic3⋊C4—61no—
173Dic3⋊C4—61no—
174Dic3⋊C4C2×D311yescharacteristic
175Dic3⋊C4C2×D311yescharacteristic
176Dic3⋊C4C2×D311yescharacteristic
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C4×Dic3C2×D311yescharacteristic
183C22×C32C2411yescharacteristic
184C2×C3×D3—21no—
185C2×C3×D3—21no—
186C2×C3×D3—21no—
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—21no—
193C3⋊Dic3—21no—
194C3⋊Dic3—21no—
195C3⋊Dic3—21no—
196C3⋊Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C4⋊Q8—91no—
204C4⋊Q8—91no—
205#126—31no—
206#126—91no—
207#126—91no—
208#126—91no—
209#125—91no—
210#125—91no—
211#124—91no—
212#123—91no—
213#123—91no—
214C22⋊Q8—31no—
215C22⋊Q8—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C23×C3—31no—
219C23×C3—31no—
220C22×D3C22×D311yescharacteristic
221C22×D3C22×D311yescharacteristic
222C3×Q8—121no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—121no—
233C2×C3×C4C22×D311yescharacteristic
234C2×C3×C4C22×D311yescharacteristic
235C2×C3×C4C22×D311yescharacteristic
236C2×C3×C4C22×D311yescharacteristic
237C2×C3×C4C22×D311yescharacteristic
238C2×C3×C4C22×D311yescharacteristic
239#67—121no—
240#67—121no—
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—121no—
254C2×Dic3C22×D311yescharacteristic
255C2×Dic3C22×D311yescharacteristic
256C2×Dic3C22×D311yescharacteristic
257C2×Dic3C22×D311yescharacteristic
258C2×Dic3C22×D311yescharacteristic
259C2×Dic3C22×D311yescharacteristic
260C4×D3—41no—
261C4×D3—121no—
262Dic6—41no—
263Dic6—121no—
264Dic6—121no—
265Dic6—121no—
266Dic6—121no—
267Dic6—122no—
268Dic6—122no—
269C2×C32H(5,𝔽2)11yescharacteristic
270C2×C322−1+411yescharacteristic
271C2×C322−1+411yescharacteristic
272C3×D3—41no—
273C3×D3—41no—
274C2×Q8—31no—
275C2×Q8—91no—
276C2×Q8—91no—
277C2×D4—91no—
278C22×C4—31no—
279C22×C4—91no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—91no—
294C4⋊C4—91no—
295C4⋊C4D3211yescharacteristic
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—31no—
300C22⋊C4—31no—
301C22⋊C4—91no—
302C22⋊C4—91no—
303C22⋊C4—91no—
304C22⋊C4—91no—
305C22⋊C4—91no—
306C42—31no—
307C42—91no—
308C42—91no—
309C22×C3—21no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3C23×D311yescharacteristic
317C22×C3C23×D311yescharacteristic
318C2×D3—21no—
319C2×D3—21no—
320C2×D3—21no—
321C2×D3—21no—
322C2×D3—21no—
323C2×D3—21no—
324C3×C4—22no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—61no—
339Dic3—21no—
340Dic3—21no—
341Dic3—21no—
342Dic3—21no—
343Dic3—21no—
344Dic3—21no—
345Dic3—62no—
346Dic3—62no—
347Dic3—61no—
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351Dic3—61no—
352Dic3—61no—
353Dic3—61no—
354Dic3—61no—
355Dic3—121no—
356Dic3—121no—
357C32#56311yescharacteristic, complemented, Sylow
358C23—31no—
359C23—31no—
360Q8—121no—
361Q8—361no—
362Q8—361no—
363D4—361no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—121no—
375C2×C4—361no—
376C2×C4C2×D3211yescharacteristic
377C2×C4C2×D3211yescharacteristic
378C2×C4C2×D3211yescharacteristic
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—121no—
383C2×C3—121no—
384C2×C3#100411yescharacteristic
385C2×C3#100711yescharacteristic
386C2×C3#100911yescharacteristic
387C2×C3#101011yescharacteristic
388C2×C3#101011yescharacteristic
389C2×C3#101011yescharacteristic
390D3—41no—
391D3—41no—
392C22—61no—
393C22—61no—
394C22—61no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22C22×D3211yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—61no—
403C4—61no—
404C4—61no—
405C4—61no—
406C4—61no—
407C4—61no—
408C4—181no—
409C4—181no—
410C4—181no—
411C4—181no—
412C3—21no—
413C3#564611yescharacteristic, complemented
414C3#571211yescharacteristic, complemented
415C2—121no—
416C2—121no—
417C2#6429811yescharacteristic
418C2#6431411yescharacteristic
419C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #64097C2semidirect
2. C3×#898C2semidirect
3. C3×#896C2semidirect
6. #63896C2semidirect
8. #63887C2semidirect
9. #63866C2semidirect
10. #63862C2semidirect
11. #63862C2semidirect
12. #63859C2semidirect
13. #63857C2semidirect
14. #63846C2semidirect
15. #63845C2semidirect
61. #898D3semidirect
64. #896D3semidirect
357. C32#563semidirect
413. C3#5646semidirect
414. C3#5712semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#640972
C3×#8982
C3×#8962
#639132
#639052
#638962
#638912
#638872
#638662
#638622
#638592
#638572
#638462
#638452
#57123
#56463